- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Diffusion of epicenters of earthquake aftershocks, Omori’s law, and generalized continuous-time random walk models

A. Helmstetter1,* and D. Sornette2,3,†

  • 1Laboratoire de Géophysique Interne et Tectonophysique, Observatoire de Grenoble, Université Joseph Fourier, Boîte Postale 53X, 38041 Grenoble Cedex, France
  • 2Laboratoire de Physique de la Matière Condensée, CNRS UMR6622 and Université de Nice–Sophia Antipolis, Boîte Postale 71, Parc Valrose, 06108 Nice Cedex 2, France
  • 3Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles, California 90095

  • *Email address: ahelmste@obs.ujf-grenoble.fr
  • †Email address: sornette@ess.ucla.edu
PDF

Phys. Rev. E 66, 061104 – Published 30 December, 2002

DOI: https://doi.org/10.1103/PhysRevE.66.061104

Abstract

The epidemic-type aftershock sequence (ETAS) model is a simple stochastic process modeling seismicity, based on the two best-established empirical laws, the Omori law (power-law decay ∼1/t1+θ of seismicity after an earthquake) and Gutenberg-Richter law (power-law distribution of earthquake energies). In order to describe also the space distribution of seismicity, we use in addition a power-law distribution ∼1/r1+μ of distances between triggered and triggering earthquakes. The ETAS model has been studied for the last two decades to model real seismicity catalogs and to obtain short-term probabilistic forecasts. Here, we present a mapping between the ETAS model and a class of CTRW (continuous time random walk) models, based on the identification of their corresponding master equations. This mapping allows us to use the wealth of results previously obtained on anomalous diffusion of CTRW. After translating into the relevant variable for the ETAS model, we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock. Specifically, we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks. The different regimes are fully characterized by the two exponents θ and μ. Our predictions are checked by careful numerical simulations. We stress the distinction between the “bare” Omori law describing the seismic rate activated directly by a mainshock and the “renormalized” Omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock, and so on. In particular, we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed Omori exponent is less than 1, because this signals the operation of the renormalization of the bare Omori law, also at the origin of seismic diffusion in the ETAS model. We present predictions and insights provided by the ETAS to CTRW mapping which suggest different ways for studying seismic catalogs. Finally, we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock, stressing the caveats and limitations of previous empirical works.

View figure in article

Comments & Replies

Comment on “Diffusion of epicenters of earthquake aftershocks, Omori’s law, and generalized continuous-time random walk models”

David Marsan and Christopher J. Bean
Phys. Rev. E 69, 063101 (2004)

Reply to “Comment on ‘Diffusion of epicenters of earthquake aftershocks, Omori’s law, and generalized continuous-time random walk models’ ”

A. Helmstetter and D. Sornette
Phys. Rev. E 69, 063102 (2004)

References (98)

  1. D. Sornette, Spontaneous Formation of Space-Time Structures and Criticality, Vol. 39 of NATO Advanced Study Institute, Series B: Physics, edited by T. Riste and D. Sherrington (Kluwer, Dordrecht, 1991), pp. 57–106.
  2. J. B. Rundle and W. Klein, Rev. Geophys. 33, 283 (1995).
  3. I. Main, Rev. Geophys. 34, 433 (1996).
  4. D. Sornette, Phys. Rep. 313, 238 (1999).
  5. D. L. Turcotte, Rep. Prog. Phys. 62, 1377 (1999).
  6. B. Gutenberg and C. F. Richter, Bull. Seismol. Soc. Am. 34, 185 (1944).
  7. F. Omori, On the aftershocks of earthquakes, J. Coll. Sci. Imp. Univ. Tokyo 7, 111 (1894).
  8. G. Ouillon, C. Castaing, and D. Sornette, J. Geophys. Res., [Solid Earth] 101B3, 5477 (1996).
  9. Y. Y. Kagan and L. Knopoff, Geophys. J. R. Astron. Soc. 62, 303 (1980).
  10. V. F. Pisarenko and D. Sornette, e-print cond-mat/0011168, Pure Appl. Geophys. (to be published).
  11. Y. Y. Kagan, Pure Appl. Geophys. 155, 537 (1998).
  12. D. Sornette, Critical Phenomena in Natural Sciences, Springer Series in Synergetics (Springer, Heidelberg, 2000).
  13. R. A. Harris, in International Handbook of Earthquake and Engineering Seismology, edited by W. H. K. Lee, H. Kanamori, P. C. Jennings, and C. Kisslinger (unpublished), Chap. 73.
  14. T. Yamashita and L. Knopoff, Geophys. J. R. Astron. Soc. 91, 13 (1987).
  15. M. W. Lee and D. Sornette, Eur. Phys. J. B 15, 193 (2000).
  16. Y. Huang, H. Saleur, C. G. Sammis, and D. Sornette, Europhys. Lett. 41, 43 (1998).
  17. T. Utsu, Y. Ogata, and S. Matsuúra, J. Phys. Earth 43, 1 (1995).
  18. M. Bath and C. F. Richter, Bull. Seismol. Soc. Am. 48, 133 (1958).
  19. G. C. Beroza and M. D. Zoback, Science 259, 210 (1993).
  20. D. P. Hill et al., Science 260, 1617 (1993).
  21. D. W. Steeples and D. D. Steeples, Bull. Seismol. Soc. Am. 86, 921 (1996).
  22. Y. Y. Kagan and D. D. Jackson, J. Geophys. Res., [Solid Earth] 103, 24 453 (1998).
  23. A. J. Meltzner and D. J. Wald, Bull. Seismol. Soc. Am. 89, 1109 (1999).
  24. D. Dreger and B. Savage, Bull. Seismol. Soc. Am. 89, 1094 (1999).
  25. A. Sornette and D. Sornette, Geophys. Res. Lett. 26, 1981 (1999).
  26. A. Helmstetter and D. Sornette, e-print http://arXiv.org/abs/ cond-mat/0109318, J. Geophys. Res. (to be published).
  27. D. Sornette and A. Helmstetter, Phys. Rev. Lett. 89, 158501 (2002).
  28. K. Mogi, J. Phys. Earth 16, 30 (1968).
  29. M. Imoto, On migration phenomena of aftershocks following large thrust earthquakes in subduction zones, Report of the National Research Center for Disaster Prevention, No. 25, 1981, pp. 29–71.
  30. J. L. Chatelain, R. K. Cardwell, and B. L. Isacks, Geophys. Res. Lett. 10, 385 (1983).
  31. F. Tajima and H. Kanamori, Phys. Earth Planet. Inter. 40, 77 (1985).
  32. F. Tajima and H. Kanamori, Geophys. Res. Lett. 12, 345 (1985).
  33. R.L. Wesson, Tectonophysics 144, 214 (1987).
  34. T. Ouchi and T. Uekawa, Phys. Earth Planet. Inter. 44, 211 (1986).
  35. J. Noir, E. Jacques, S. Bekri, P. M. Adler, P. Tapponnier, and G. C. P. King, Geophys. Res. Lett. 24, 2335 (1997).
  36. E. Jacques, J. C. Ruegg, J. C. Lepine, P. Tapponnier, G. C. P. King, and A. Omar, Geophys. J. Int. 138, 447 (1999).
  37. P. A. Rydelek and I. S. Sacks, Geophys. Res. Lett. 28, 3079 (2001).
  38. A. Nur and J. R. Booker, Science 175, 885 (1972).
  39. K. W. Hudnut, L. Seeber, and J. Pacheco, Geophys. Res. Lett. 16, 199 (1989).
  40. J. Dieterich, J. Geophys. Res., [Solid Earth] 99, 2601 (1994).
  41. Y. Y. Kagan and L. Knopoff, Science 236, 1563 (1987).
  42. Y. Ogata, Am. Stat. 83, 9 (1988).
  43. Y. Yamanaka and K. Shimazaki, J. Phys. Earth 38, 305 (1990).
  44. G. Drakatos and J. Latoussakis, Journal of Seismology 5, 137 (2001).
  45. Z. Guo, and Y. Ogata, J. Geophys. Res., [Solid Earth] 102, 2857 (1997).
  46. A. Helmstetter, Geophys. Res. Lett. (to be published).
  47. K. R. Felzer, T. W. Becker, R. E. Abercrombie, G. Ekström, and J. R. Rice, J. Geophys. Res., [Solid Earth] (to be published).
  48. M. Bath, Tectonophysics 2, 483 (1965).
  49. R. Console, A. M. Lombardi, M. Murru, and D. Rhoades, J. Geophys. Res. (to be published).
  50. Y. Ogata, Tectonophysics 169, 159 (1989).
  51. Y. Ogata, J. Geophys. Res., [Solid Earth] 97, 19 845 (1992).
  52. Y. Ogata, Pure Appl. Geophys. 155, 471 (1999).
  53. Y. Ogata, J. Geophys. Res., [Solid Earth] 106, 8729 (2001).
  54. Y. Y. Kagan, Geophys. J. Int. 106, 135 (1991).
  55. Y. Y. Kagan and D. D. Jackson, Geophys. J. Int. 143, 438 (2000).
  56. R. Console and M. Murru, J. Geophys. Res., [Solid Earth] 166, 8699 (2001).
  57. E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965).
  58. S. Krishnamurthy, A. Tanguy, P. Abry, and S. Roux, Europhys. Lett. 51, 1 (2000).
  59. Y. Ogata, Ann. Inst. Stat. Math. 50, 379 (1998).
  60. C. H. Scholz and B. B. Mandelbrot, Fractals in Geophysics (Birkhäuser Verlag, Boston, 1989).
  61. A. Sornette, P. Davy, and D. Sornette, J. Geophys. Res. [Solid Earth] 98, 12 111 (1993).
  62. C. C. Barton and P. R. La Pointe, Fractals in the Earth Sciences (Plenum Press, New York, 1995).
  63. K. Nanjo and H. Nagahama, Pure Appl. Geophys. 157, 575 (2000).
  64. G. Ouillon and D. Sornette, Geophys. Res. Lett. 23, 3409 (1996).
  65. D. Hamburger, O. Biham, and D. Avnir, Phys. Rev. E 53, 3342 (1996).
  66. M. Eneva, Geophys. J. Int. 124, 773 (1996).
  67. O. Malcai, D. A. Lidar, O. Biham, and D. Avnir, Phys. Rev. E 56, 2817 (1997).
  68. A. Helmstetter, D. Sornette, and J.-R. Grasso, e-print http://arXiv.org/abs/ cond-mat/0205499, J. Geophys. Res. (to be published).
  69. E. W. Montroll and H. Scher, J. Stat. Phys. 9, 101 (1973).
  70. H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455 (1975).
  71. V.M. Kenkre, E. W. Montroll, and M. F. Shlesinger, J. Stat. Phys. 9, 45 (1973).
  72. M. F. Shlesinger, J. Stat. Phys. 10, 421 (1974).
  73. G. H. Weiss, Aspects and Applications of Random Walk (North-Holland, Amsterdam, 1994).
  74. B. D. Hughes, Random Walks and Random Environments (Oxford University Press, New York, 1995).
  75. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
  76. G. Margolin and B. Berkowitz, J. Phys. Chem. 104, 3942 (2000).
  77. B. Berkowitz and H. Scher, Transp. Porous Media 42, 241 (2001).
  78. S. Redner, Am. J. Phys. 58, 267 (1990).
  79. H. Scher, H. G. Margolin, R. Metzler, J. Klafter, and B. Berkowitz, Geophys. Res. Lett. 29, 5 (2002).
  80. J. W. Kirchner, X. Feng, and C. Neal, Nature (London) 403, 524 (2000).
  81. E. Barkai, R. Metzler, and J. Klafter, Phys. Rev. E 61, 132 (2000).
  82. A. M. Mathai and R. K. Saxena, The H-function with Applications in Statistics and Other Disciplines (Wiley Eastern Limited, New Delhi, 1978).
  83. H. E. Roman and P. A. Alemany, J. Phys. A 27, 3407 (1994).
  84. E. Barkai, Phys. Rev. E 63, 046118 (2001).
  85. C. Budde, D. Prato, and M. Re, Phys. Lett. A 283, 309 (2001).
  86. M. Kotulski, J. Stat. Phys. 81, 777 (1995).
  87. E. Barkai, e-print cond-mat/0108024.
  88. A. I. Saichev and G. M. Zaslavsky, Chaos 7, 753 (1997).
  89. C. M. Bender and S. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  90. G. H. Weiss and P. P. Calabrese, Physica A 234, 443 (1996).
  91. I. M. Sokolov, A. Blumen, and J. Klafter, e-print cond-mat/0107632.
  92. I. M. Sokolov, Phys. Rev. E 63, 011104 (2001).
  93. C. M. Bender, S. Boettcher, and P. N. Meisinger, Phys. Rev. Lett. 75, 3210 (1995).
  94. B. E. Shaw, Geophys. Res. Lett. 20, 907 (1993).
  95. D. Marsan, C. J. Bean, S. Steacy, and J. McCloskey, J. Geophys. Res. [Solid Earth] 105, 28 (2000).
  96. D. Marsan, C. J. Bean, S. Steacy, and J. McCloskey, Geophys. Res. Lett. 26, 3697 (1999).
  97. D. Marsan and C. J. Bean, Geophys. J. Int. (to be published).
  98. O. Sotolongo-Costa, J. C. Antoranz, A. Posadas, F. Vidal, and A. Vásquez, Geophys. Res. Lett. 27, 1965 (2000).

Outline

Information

Phys. Rev. E 66, 061104– Published 30 December, 2002
  • Received 25 March 2002

© 2002 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation